The Helmholtz decomposition of a $BMO$ type vector field in a slightly perturbed half space

We introduce a space of $L^2$ vector fields with bounded mean oscillation whose normal component to the boundary is well-controlled. We establish its Helmholtz decomposition in the case when the domain is a perturbed $C^3$ half space in ${\mathbf{R}}^n$ $(n\ge 3)$ with small perturbation.